Start by browning the butter in a saucepan over medium heat until it turns a golden brown color, stirring frequently. Remove from heat and let cool slightly.
In a large mixing bowl, whisk together the oats, flour, baking soda, cinnamon, nutmeg, and salt. Set aside.
In another bowl, combine the brown sugar and granulated sugar. Add the cooled brown butter, eggs, vanilla, and molasses. Mix until smooth.
Gradually add the dry mixture to the wet ingredients, stirring gently until just combined. Avoid overmixing.
Cover the bowl with plastic wrap and chill the dough in the refrigerator for at least 30 minutes.
Preheat oven to 350°F (175°C). Scoop tablespoon-sized portions onto baking sheets lined with parchment paper. Bake for about 10 minutes until lightly golden.
In a clean bowl, combine sifted powdered sugar with milk or cream and vanilla. Whisk until smooth, adjusting consistency as needed.